COVERS FOR S-ACTS AND CONDITION (A) FOR A MONOID S

Abstract

A monoid S satisfies Condition (A) if every locally cyclic left S-act is cyclic. This condition first arose in Isbell's work on left perfect monoids, that is, monoids such that every left S-act has a projective cover. Isbell showed that S is left perfect if and only if every cyclic left S-act has a projective cover and Condition (A) holds. Fountain built on Isbell's work to show that S is left perfect if and only if it satisfies Condition (A) together with the descending chain condition on principal right ideals, MR. We note that a ring is left perfect (with an analogous definition) if and only if it satisfies MR. The appearance of Condition (A) in this context is, therefore, monoid specific. Condition (A) has a number of alternative characterisations, in particular, it is equivalent to the ascending chain condition on cyclic subacts of any left S-act. In spite of this, it remains somewhat esoteric. The first aim of this paper is to investigate the preservation of Condition (A) under basic semigroup-theoretic constructions. Recently, Khosravi, Ershad and Sedaghatjoo have shown that every left S-act has a strongly flat or Condition (P) cover if and only if every cyclic left S-act has such a cover and Condition (A) holds. Here we find a range of classes of S-acts $\mathcal{C}$ such that every left S-act has a cover from $\mathcal{C}$ if and only if every cyclic left S-act does and Condition (A) holds. In doing so we find a further characterisation of Condition (A) purely in terms of the existence of covers of a certain kind. Finally, we make some observations concerning left perfect monoids and investigate a class of monoids close to being left perfect, which we name left$\mathcal{IP}$a-perfect.

Perfection for semigroups

“…The following result is inspired by Result 1.2 in [9], by Lemma 1.3 in [10] and also by Lemma 2.2 and Theorem 5.2 in [2]. Among other things, it shows that Condition (A) can be given a description that does not refer to acts but only uses the elements of S. Theorem 4.3.…”

“…Remark 4.4. Condition (5) in Theorem 4.3 appears first in Lemma 2.2 of [2], but a similar condition was used in [10]. It is kind of interesting that this condition can be formulated in topological terms as follows.…”

“…In particular, he proved that a monoid is left perfect if and only if all of its weakly flat (in the sense of being pullback flat) unitary left acts are projective. Subsequently, a number of other papers related to perfect monoids or pomonoids have appeared (e.g., [2,8,[10][11][12]20]).…”

“…Cf [2,. Definition 4.1] We say that a semigroup S is right ICperfect if every unitary act A S has a cover f : B S → A S , where B S is unitary, and indecomposable components of B S are cyclic.…”

Covers of Finitely Generated Acts over Monoids

“…In Section 5 we study Enochs' notion of cover in the category of acts over monoids and focus on X -precovers, where X is a class of S-acts closed under isomorphisms. Basic results on covers of acts over monoids can be found in [3,4,6,11].…”